Present day processing systems and methods have to deal with large amounts of data. To facilitate the handling of the large amounts of data, data compression is often used. The data may be compressed using some specific compression methods; such as for example using consecutive data difference methods (delta-modulation). The data is operated on and/or stored in the compressed mode. Subsequentially, the data is once again expanded for its actual use; forming an image, for example.
The compression ratio (CR) of the compressed data, that is the ratio of the quantity of non-compressed data to the quantity of compressed data, is a quantitative measurement of the efficiency of the data compression system.
The CR is limited by the variability of the data, among other things. Further constraints are imposed by the fidelity requirements of the system. For example, very high CRs can be achieved if only a little variability is accomodated; e.g., the whole data set can be replaced by its mean;--a single number. The CR would then be equal to the original number of data units (numbers), say 262, 144 for a 512.times.512 matrix of image data. However, this method of compression incurs a huge loss of fidelity as the original data usually does have some variability containing important information which is lost in the compression method of the above example.
Many compression methods use "predictors" of some kind. Predictors use a small number of parameters and a small input of data points, to "predict" the rest of the data. Some predictors are "running" predictors whose input is the last few data points and whose output is the next data point. An example of a "predictor" method is the "difference" or "delta modulation" method, which "predicts" that each data point is followed by another data point with the same value. These "predictors" must be completed with "correctors", measuring and correcting the predictor error. Using a suitable predictor can greatly decrease the variability of the data, and thus enable much higher CRs. However, this improvement is limited by random inaccuracies, which are commonly called "noise". Being random they are unpredictable, putting a lower limit to the variability, and consequently a top limit to the possible CR with a given fidelity. According to the second law of thermodynamics, it is impossible to completely separate noise from meaningful (and predictable) variations. However, in the general case of the frequency content (power spectrum) of the data, the higher the frequency, the higher the relative noise content, until at some limit frequency the noise dominates. Therefore all data at frequencies higher than this limit can be viewed as pure noise. This limit is never higher than the Nyquist frequency, which is determined by the data sampling rate (it is the highest frequency that can be correctly sampled). Based on this property, sometimes low-pass filters (smoothings) are used to reduce the high frequency content in the data, thus improving the predictability and hence the CR in predictor methods.
However, reducing the noise by filtering often results in deleting actual data from the system, therby decreasing the fidelity of the system. A compromise therefore, has to be reached between noise filtering and the fidelity of the system.
The "compromise" is often set to the side of fidelity; therby, reducing the compression ratio; i.e., the efficiency of the compression. This is true for example in medical imaging where the fidelity of the final data can make the difference between a proper diagnosis of a patient's problems and an incorrect diagnosis. Therefore, the physicians and clinicians tend to arrange the systems to prevent deletion of actual signals. Thus, in the past, most smoothing filters that were used prior to compressing the data, if any, were limited by the desire to maintain the fidelity in order to reproduce all of the data when the smoothed and compressed data is expanded.
It must be noted that the natural variability of the data and the "signal to noise ratio" (SNR) vary with application, data acquisition system and specific data. Therefore, the compromise chosen is highly dependent on the above variables as well as personal preferences. The medical imaging systems usually have low SNR, thus making high fidelity compression methods inefficient.
The large amounts of data handled by present equipment such as a medical imaging equipment warrants the use of systems which will make the data compression much more efficient and effective without significantly reducing the fidelity.
One apparent way of balancing the opposing demands of high CR and high fidelity is to "undo" the effect of the smoothing after expanding the data. This is sometimes called "restoration". As before, the second law of thermodynamic prevents complete restoration. For example, the use of sharpening high pass filters (to undo the effects of smoothing) tends to increase the noise more than the data; and, in medical images, where the noise is relatively large to begin with, this is unacceptable and may even lead to numerical instability. The algebraic solution of N simultaneous equations (N being the number of pixels in the image) is both impossibly long and numerically instable. It is well known that the time required for solving a set of N simultaneous algebraic equations increases as N to the 4th power and in an image of 512.times.512 pixels we have 262, 144 equations making the problem virtually insoluble using the presently available equipment. Thus, an apparent way of balancing the opposing goals of high CR and high fidelity appears to be blocked.
A definition is supplied here of a seperable filter to aid in understanding the description of the invention:
A separable filter is a filter that can be effected by operating its two parts consecutively. (The order does not matter for "linear" filters).
The separability of filter F is mathematically denoted as: EQU F=F1*F2=[F2*F1]
The mathematical notation for operating the separable filter F on a data set X is: EQU F1*(F2*X)=(F1*F2)*X=F*X=[(F2*F1)*X=F2*(F1*X)]
The part in square brackets denotes symmetry with respect to order (cummulativity).